The Calkin algebra is not countably homogeneous

Abstract

We show that the Calkin algebra is not countably homogeneous, in the sense of continuous model theory. We furthermore show that the connected component of the unitary group of the Calkin algebra is not countably homogeneous. Motivated by their study of extensions of C∗-algebras, Brown, Douglas and Fillmore asked whether the Calkin algebra has a K-theory reversing automorphism and whether it has outer automorphisms at all ([4, Remark 1.6 (ii)]). By [16] and [8] the answer to the latter question is independent from ZFC. In particular, since inner automorphisms fix K-theory, a negative answer to the former question is relatively consistent with ZFC. It is not known whether the existence of a K-theory reversing automorphism of the Calkin algebra is relatively consistent with ZFC. All known automorphisms of the Calkin algebra ([16] and [8, §1]) act trivially on its K-theory, as they are implemented by a unitary on every separable subalgebra of the Calkin algebra. A scenario for using Continuum Hypothesis to construct a K-theory reversing automorphism of the Calkin algebra on separable Hilbert space, denoted Q, was sketched in [11, §6.3] and in [9, §7.1]. The following theorems demonstrate that this strategy does not work and suggest that question of the existence of such automorphism is even more difficult than previously thought (for terminology see below and [11] or [10]). Theorem 1. The Calkin algebra Q is not countably homogeneous, and this is witnessed by a quantifier-free type. Theorem 2. The group U0(Q) of Fredholm index zero unitaries in Q is not countably homogeneous. Our theorems give negative answers to [10, Questions 5.2 and 5.7] and a novel obstruction to countable saturation of Q. In [10, Question 5.1] it was asked whether all obstructions to (quantifier-free) countable saturation of Q are of K-theoretic nature. The obstruction given in our results is finer than the Fredholm index, but it is K-homological and therefore ultimately K-theoretical. In addition, the obstruction given in Theorem 1 is quantifier-free and one given in Theorem 2 appears to have little to do with the Fredholm index. It should be noted that one of the key ideas, using Ext(M2∞), is due to N.C. Phillips, and it was already used in the proof of [10, Proposition 4.2]. Date: January 31, 2016.